In this work, we propose to represent the coordinates of the input geometry with a positional encoding based on the eigenfunctions of the Laplace-Beltrami operator of the manifold. In this way, points that are close in the surface will remain close in the positional encoding space. Next, the input of the neural network will be the value of a finite number of the eigenfunctions evaluated at a point in the surface and the output will remain the physical quantity that we are modeling with PINNs. Positional encoding have shown great success in improving the capabilities of neural networks and are used in transformers [1]}, neural radiance fields [2]} and PINNs [3]} to name a few. For our positional encoding, the Laplace-Beltrami eigenfunctions can be approximated numerically for any shape, e.g., with the use of the finite element method. By approximating the eigenfunctions we lose the ability to use automatic differentiation to compute the operators of the partial differential equations. However, we show that common operators such as the gradient and the Laplacian can be efficiently computed with finite elements applied to the output of the neural network. We demonstrate the capabilities of the proposed method by testing different geometries, such as a coil, a heat sink and a bunny and different physics, such as the Eikonal equation and heat transfer.
